Critical Thresholds in Euler-Poisson Equations
| dc.creator | Engelberg, Shlomo | |
| dc.creator | Liu, Hailiang | |
| dc.creator | Tadmor, Eitan | |
| dc.date | 2001-12-03 | |
| dc.date.accessioned | 2026-07-07T04:44:56Z | |
| dc.date.available | 2026-07-07T04:44:56Z | |
| dc.description | We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -- the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic, ${O}(1)$ critical threshold. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problem with or without various forcing mechanisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time,the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock discontinuity except for the non-generic case of increasing initial profile, $u_0' \geq 0$. In contrast, we show that the corresponding one dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initial $(ρ_0,u_0)$- configuration satisfies $u_0'\geq -\sqrt{2kρ_0}$, allowing a finite, critical negative velocity gradient. As is typical for such nonlinear convection problems one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena. | |
| dc.identifier | https://arxiv.org/abs/math/0112014 | |
| dc.identifier | http://arxiv.org/abs/math/0112014 | |
| dc.identifier | Indiana University Mathematics Journal 50(1) 2001, 109-157 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62797 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Q35; 35B30 | |
| dc.title | Critical Thresholds in Euler-Poisson Equations | |
| dc.type | text |